Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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2answers
48 views

Isn't $ n(n-1)(n-2)…(n-m+1) $ a factorial already?

Let $ m \ge 1 $ and $ n \ge 1 $ be integers Let $A$ be a set of size $m$ Let $B$ be a set of size $n$ How many one-to-one functions $f: A \rightarrow B$ are there? skipped stuff $$ ...
0
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1answer
31 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
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1answer
17 views

Series increasing or decreasing with factorials

I have been working on some homework for calc 3 and my prof has put a couple sequences in which we must find if they are increasing or decreasing with factorials in them. I've googled and there are ...
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1answer
45 views

Computing $\mathrm{gcd} (100!, 3^{100})$

I am trying to compute $\mathrm{gcd}(100!,3^{100})$. I am still not sure how to reach an answer but I feel that Wilson's Theorem (i.e., $(p-1)!\equiv -1 \bmod p, p$ prime) and Fermat's Little theorem ...
0
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3answers
76 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
3
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1answer
41 views

Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any ...
1
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2answers
33 views

Factorial formula problem [duplicate]

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
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1answer
42 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
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1answer
103 views

Finding the largest factorial of only three digits

I am using the following Python code to compute the above, but no results up to 16000!: ...
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2answers
56 views

$x!=y^n$ for $x,y \neq 0,1$

A straightforward problem (find all integers such that $m!+3=n^2$) led me into thinking about the integers for which: $$x!=y^2$$ is true. I argued that other than the trivial case ($x!=1$) that this ...
0
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2answers
140 views

Inequality $(n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n$

Prove that $$ (n!)^2\le \left[\frac{(n+1)(n+2)}{6}\right]^n $$ holds for all $n\in\mathbb{Z^+}$. I tried induction but there's no obvious way to go from $n$ to $n+1$.
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2answers
130 views
+50

Proving an identity involving factorials

I have stumbled upon the following statement and have verified it computationally for many $n$ (up to n=500, it took a long time for my computer to do out all of the math), yet I have no idea how to ...
0
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0answers
17 views

Function to define how combinations N items can be organized with a certain condition

This is not a factorial only problem If I have 5 items and I wanted to know how many possible ways they could be arranged, the answer is 5! or 120. However my situation is I need to know how many ...
0
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3answers
49 views

Simplifying factorials

I apologise for a repost, but my rep is not high enough to ask in a comment. But, in this question Simplify sum of factorials with mathematical induction I am confused how: $$(n+1)!-1+(n+1)(n+1)! = ...
3
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3answers
71 views

Proving $\frac{((2n)!)^2(i)!(j)!}{((n)!)^2(2i)!(2j)!}$ is an integer

I have verified this for many values of $n$, but I have no idea how to prove it. Does anyone know how I could go about showing that: $$\frac{((2n)!)^2(i)!(j)!}{((n)!)^2(2i)!(2j)!}$$ is an integer when ...
2
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1answer
23 views

Is there a way to express $(n-i)!(n-j)!(2i)!(2j)!$ in terms of $n$ and $r=i+j$?

I have been attempting to simplify the double sum: $$\sum_{i=0}^n \sum_{j=0}^n \frac{(-1)^{i+j} (2i+2j)!}{(n-i)!(n-j)!(2i)!(2j)!2^{i+j}(i+j)!}$$ And so what I am attempting to do is rewrite it in ...
1
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0answers
43 views

Limit of $\frac{1}{n}\log({n\choose np})$ without using Stirling's formula

I am trying to evaluate the following limit: $$ \forall p\in(0,1) ,\lim_{n\rightarrow \infty} \frac{1}{n}\log{n\choose \lfloor np \rfloor} =H(p),$$ where $\lfloor x\rfloor$ means the greatest integer ...
4
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3answers
56 views

Proving $ n! \geq 2^{n-1} $

Prove that $$ n! \geq 2^{n-1}$$ for $n \geq 1$. My initial solution by induction goes like this. For $n = 1 : 1 \geq 1 $. Assuming that $$ n ! \geq 2^{n-1}.$$ Then for $n+1$, $$ (n+1)! = ...
2
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3answers
54 views

Evaluate: $\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$

Evaluate: $$\sum_{k=2}^n\frac {n!}{(n-k)!(k-2)!}$$ Attempt $S_2=\frac {n!}{(n-2)!}$ $S_3=\frac {n!}{(n-3)!}$ $S_4=\frac {n!}{2(n-4)!}$ $\vdots$ $S_{n-1}=\frac {n!}{1!(n-3)!}$ $S_n=\frac ...
3
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4answers
58 views

Showing $\binom{2n}{n} = (-4)^n \binom{-1/2}{n}$

Is there a proof for the following identity that only uses the definition of the (generalized) binomial coefficient and basic transformations? Let $n$ be a non-negative integer. $$\binom{2n}{n} = ...
1
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1answer
31 views

Is there a closed form for this sequence?

I'm trying to find a closed form for the following sequence: $a$ $a(a-1)$ $a(a-1)(a-2)$ $a(a-1)(a-2)(a-3)$ The problem is, $a=\frac{1}{2}$. If it were some whole number, then I'd use ...
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2answers
72 views

Question about Binomial Sums [duplicate]

Prove that for any $a \in \mathbb{R}$ $$\sum_{k=0}^n (-1)^{k}\binom{n}{k}(a-k)^{n}=n!$$ I rewrote the sum as $$\sum_{k=0}^n \left((-1)^{k}\binom{n}{k} \sum_{i=0}^n (-1)^{i}a^{n-i} k^{i} ...
4
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2answers
65 views

How to calculate what this power series converges against? (double factorials)

I'm working on my physics master course homework and I'm given the following equation out of nowhere: $\displaystyle{ 1 + \sum_{n\ =\ 1}^{\infty}{z^n\left(\, 2n - 1\,\right)!! \over 2n!!} ={1 \over ...
3
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0answers
123 views

Sum problem involving Factorials

got the following problem to prove for $n \in \mathbb{N}$ and $1 \leq i \leq n$: \begin{equation} \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! ...
1
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0answers
19 views

What condition does a recursive function have to fulfill to be well defined?

What condition does a recursive function have to fulfill to be well defined? Provide a well-defined recursive definition of the factorial of a number. Modify the definition so that is no longer ...
1
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1answer
126 views

Need help finding a closed form for complicated sum

I'm trying to find a closed form expression for the following sequence: $$a_n=\sum_{i=1}^{n}\frac{(n-1-i+d)!}{(n-2i)!(i)!}=\sum_{i=1}^{\frac{n}{2}}\frac{(n-1-i+d)!}{(n-2i)!(i)!}$$ Where $n$ and $d$ ...
3
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1answer
24 views

How to simplify a fraction involving factorials

I have following term: $$\frac{\frac{3^{2k+2}}{(2k+2)!}}{\frac{2^{2k}}{(2k)!}}=\frac{3^{2k+2}\cdot(2k)!}{(2k+2)! \cdot 3^{2k}}=9\cdot\frac{(2k)!}{(2k+2)!}$$ I know that you can simplify even further ...
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0answers
138 views

Sum concerning Catalan triangle and binomials

I am trying to prove the following relation. For $k \in \mathbb{N},k \geq 2$ and $i=1,\ldots,k-1$ \begin{equation} {2k-2-i \choose k-1-i}=\sum \limits_{l=0}^{k-1-i} 2^{2l} T(k-2-l,k-1-i-l)-2 \sum ...
3
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4answers
102 views

Solving for $r$ in ${12\choose{r}}=924$

I can solve the equation $_{12}C_r=924$ fairly easily by guess and test because there are so few possible $r$ values, but is there a clean way to solve an equation of this format algebraically? I ...
2
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1answer
52 views

Limit of $\lim \limits_{x \rightarrow \infty}\frac{(x!)^n}{(ax)!}$

For a given $n \geq 1$ $\hspace{2mm}(n \in \mathbb{R})$, I know that $$\lim \limits_{x \rightarrow \infty}\frac{(x!)^n}{(ax)!},$$ only exists and it is equal to zero if $a \geq n.$ However, I cannot ...
1
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3answers
77 views

convergence of $\sum_{n=2}^{\infty}\frac{(n+1)!(n+1)^{n-1}}{n^{2n}}$

$$\sum_{n=2}^{\infty}\frac{(n+1)!(n+1)^{n-1}}{n^{2n}}$$ I used the Cauchy test and it lead me to $\frac{\sqrt[n]{n!}}{n^2}$. But I can't tell what is the limit of this. I tried the Squeeze theorem: ...
1
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0answers
24 views

Permutations and combinations - divisibility of a factorial

Question: Find the largest value of $n$ for which $125!$ is divisible by $6^n$ Approach: I tried to find all the numbers which were a multiple of 6. The number of times such divisors occurred ...
1
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2answers
41 views

Number of zeroes at end of factorial

Question: How many zeroes will there be at the end of $(127)!$ Approach: Considering the fact that when two numbers ending in $x$ and $y$ zeroes are multiplied, the resulting number contains $x+y$ ...
2
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0answers
98 views

How many zeroes would be there at the end of $11^{(5!)!}-1$?

$$11-1=10 \\ 121-1=120 \\ 1331-1=1330$$ Now it can be seen that the tens digit increases by 1 at each increment of exponent. So, only in case of $11^{10}$ the tens digit is zero and the units digit ...
5
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3answers
177 views

For all $n>2$ there exists a prime number between $n$ and $ n!$

How to prove that there exists a prime number between $n$ and $ n!$, for all $ n> 2$? (Bertrand's postulate gives a much better bound, but this question is about obtaining a self-contained ...
1
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1answer
59 views

general formula for permutations with some ordering

Assume I want all permutations of a set of numbers with certain numbers must go before others. Similar to this question but I'm looking for a more general formula. For example the set ...
0
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1answer
43 views

Proving that $\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}]$

I'm trying to prove that $$\sum_{i=0}^{n-p} \frac{i!}{(p+i)!} = \frac{1}{p-1}\left[\frac{1}{(p-1)!}-\frac{(n-p+1)!}{n!}\right]$$ for $p,n \geq 2$, $p, q \in \mathbb{N}$. I'm trying to use induction ...
1
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3answers
333 views

Other representations of factorial

I have a little question: which other representation of factorial $n!$ without using the factorial? Is there any definition of factorial as a series? or any other way? Thanks in advanced.
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4answers
143 views

How many digits are there in 100!? [closed]

How many digits are there in 100 factorial? How does one calculate the number of digits?
3
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3answers
76 views

Writing number as sum of reciprocals of factorial

Given a real number $r>0$. Is there a way to determine whether $r$ can be written as a (possibly infinite) sum of distinct terms of the form $1/n!$? For example, if we want to determine whether ...
1
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3answers
86 views

Factorial inequality $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n$ using induction

I want to show $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n.$ Assume that it holds for some positive integer $k\geq 1$ and we will prove, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
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1answer
289 views

Help with difficult telescoping series question

Evaluate $$\frac3{1!+2!+3!}+\frac4{2!+3!+4!}+\ldots+\frac{2012}{2010!+2011!+2012!}\;.$$ I see that the question is telescoping, but I don't know how to break it down into a form similar to that of ...
0
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1answer
27 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} ...
1
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3answers
34 views

What is the growth rate of the logarithm of the factorial sequence?

I'd like to know the space complexity of storing bit string representations of the numbers in the factorial sequence (as in a memoized factorial function). So each number $f_i=i!$ in $i=0 \cdots n$ ...
3
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2answers
106 views

Summing of factorials to produce perfect cubes

I was playing around with factorials the other day, and I realized that $4!+5!$ is a perfect square. Perplexed by this result, I started looking for other pairs of factorials that produce a perfect ...
3
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0answers
65 views

Are there infinite primes of the form n!+1? [duplicate]

Are there infinite primes of the form n!+1 ? I searched the internet but couldn't find the answer.
0
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3answers
33 views

The only solution of the equation ${72_8!}/{18_2!}=4^x$ is $x=9$

Problem and Definitions If $n_a!:=n(n-a)(n-2a)(n-3a)\ldots(n-ka):n>ka$, how should I go about solving this?: $$\dfrac{72_8!}{18_2!}=4^x$$ Attempt ...
3
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4answers
194 views

Formula for $1! \times 2! \times \cdots \times n!$?

Are there any useful forms for the expression $1!\cdot 2!\cdot 3!\cdot ...\cdot n!$? I'm trying to solve a problem that involves this expression and thought it might help to find a more "workable" ...
0
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2answers
37 views

Conditional Probability in Poker

I'm thinking of a ten person Texas hold'em game. Each person is dealt 2 cards at the start of the game. The question is: GIVEN that you have been dealt 2 hearts (Event B), what is the probability ...
1
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1answer
46 views

What non-integer number has the smallest factorial? [duplicate]

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally. ...